The corder ( correlation and order analyzor package ) was made, on July 17 2017, with the aim of analysis structures and its properties from the trajectory of the VASP's first-princiles molecular dynamics ( FPMD ) calculation.
The required files are follows:
| file | role |
|---|---|
| XDATCAR | trajectory of the VASP's FPMD calculation |
| param.in | parameters that user-controllable for analyze XDATCAR |
--
The corder performs with depends on these packages:
| file | role |
|---|---|
| includes | some auxiliary functions |
| linalg | linear algebra functions that overwritting its operators |
| voro++ | package for calculate Voronoi diagram |
| qhull | package for create convex fulls by Qhull algorithm |
Note that, some lines were added into the voro++ source code for perform the corder and draw Voronoi diagrams which is formatted to the POV-Ray style.
--
Download and install:
git clone [email protected]:Cetus-K/corder.git
cd ./corder
./INSTALL.sh
Note that, the install prefix is set to current directory. If you specify the install location, change this line:
export PREFIX=/path/to/install/location
The corder supports several geometric algorithms as follows:
| function | calculates |
|---|---|
gofr |
Pair correlation function |
sofq |
Static structure factor |
dfc |
Mean-squared displacement for diffusion coefficient, and averaged velocity |
lboo |
Local bond-orientational order parameter |
bacf |
Bond-angle correlation function |
pofqw |
Bond-orientational probability distribution |
bofree |
Bond-orientational ( Landau ) free energy coefficient |
csform |
Distrubution of cluster's sharing formations w. r. t. its sharing type |
povoro |
Make trajectory animtion of Voronoi cell |
--
Each functions are represented by the total, partial and "specified"-partial component. For example, put a function , the each component are expressed as follows:
| symbol | component |
|---|---|
| total | |
| partial by |
|
| partial by |
Since this term, I abbreviate the symbol of each component.
--
Correlations between each two-atoms-pair related to positions are directly calculated by
.
The indicates radius between two atoms with unit of
, and
means the number of other atoms at the distance
from the center atom.
--
The spectrums of the interference between neutrons in the structure are calculated by
,
This function is derived by integrate, equal to the Fourier transform of, the pair correlation function as liquid state. Where the means the norm of the wave vector with unit of
. The
, and the average atomic density of the structure
. The angular bracket indicates the time averaging related to the instantaneous variable
.
--
The mean-squared displacement , defined by its each instantaneous location from
to
, derives its diffusion coefficient
using the Einstein equation:
.
--
The ligand forms a geometric shape for an atom, which can be represented by the identical values that is likes the spectrum as follows:
,
where,
.
The coefficients in the summation of is the Wigner
symbols, and the
is the normal vector of the surface
.
For robust calculating of the cluster symmetries, there is the method that takes summation with Voronoi facet area
, and the facets
, as a weighting parameter.
These values
and
are rotationally invariant so it give us the interpretation for the distinguish cluster symmetry in any coordinates and also perspectives.
--
The cluster symmetries are correlated to the other symmetries, and the sustainability directly reflects its order as the correlation length. Also, the length can statistically estimate that the clusters are connected by being tied in a row. The definition is follows:
,
where,
.
This angular bracket takes ensamble average, and swapping the order -summation derives follows by generalized addition theorem, this package actually evaluate it:
,
where indicates the inner product between bonds which parallel with each normal vector. This returns the value of the
-th order correlation related to angular
.
--
Structures have a "geometric" free energy caused by cluster symmetries, and bond-orientational order describes that characteristics as the order parameter of the Landau theory. This function calculates internal and primary external free energy:
where,
The full Landau free energy is constructed from two more additional terms:
The first and second of additional term are secondary external free energy and periodic external free energy. I assume that the first term is usually small value and second term is always zero due to the -point sampling during the FPMD calculation.
--
Clusters are linked to each other in the system as various shape, and then there are some sharing part on the clusters. These sharing forms are categorized as follows:
| form | sharing part |
|---|---|
| isolated | nothing |
| corner sharing | a vertex |
| edge sharing | an edge |
| surface sharing | a surface |
| bicap sharing | some part of volume |
This function returns the distribution of these amount.